Let $\mathbf{Set}$, $\mathbf{Grp}$, and $\mathbf{Grp}^{\rm conj}$ denote the categories of sets and functions, groups and homomorphisms, and groups and homomorphisms up to conjugation, respectively. (Thus the fundamental group is a functor from path-connected spaces to the latter.)

Let $X$ be the forgetful functor from $\mathbf{Grp}$ to $\mathbf{Set}$, $A$ the obvious functor from $\mathbf{Grp}$ to $\mathbf{Grp}^{\rm conj}$, and $L$ the functor from $\mathbf{Grp}^{\rm conj}$ to $\mathbf{Set}$ sending a group to its set of conjugacy classes. Then is $L$ the left Kan extension of $X$ along $A$?